Bifurcations of the Spatial Central Configurations in the 5-Body Problem

On the central configurations in the spatial 5-body problem with four equal masses

“…This explains why the work of this paper completes the study of Alvarez-Ramírez et al (2008). Thus the work done in Alvarez-Ramírez et al (2008) and the one in the present paper provide a skeleton of the families of symmetric central configurations of the spatial 5-body problem with four equal masses and their bifurcations. The results that we have obtained are represented in Fig.…”

“…More precisely, we will continue numerically all the symmetric central configurations of the spatial 5-body problem with the five masses equal to 1 to the restricted spatial (4 + 1)-body problem with four masses equal to 1 and a fifth infinitesimal mass, and vice versa; that is, we vary the non equal mass from 1 to 0, and viceversa. This study completes the one presented in Alvarez-Ramírez et al (2008) where the authors continue numerically the symmetric central configurations from the spatial 5-body problem with the five masses equal to 1 to the restricted spatial (1 + 4)-body problem with four infinitesimal masses equal to m = 0 and a fifth mass equal to 1. Note that the study in Alvarez-Ramírez et al (2008) is equivalent to study the symmetric central configurations of the 5-body problem with four masses equal to 1 varying the fifth mass 1/m from 1 to infinity.…”

“…This study completes the one presented in Alvarez-Ramírez et al (2008) where the authors continue numerically the symmetric central configurations from the spatial 5-body problem with the five masses equal to 1 to the restricted spatial (1 + 4)-body problem with four infinitesimal masses equal to m = 0 and a fifth mass equal to 1. Note that the study in Alvarez-Ramírez et al (2008) is equivalent to study the symmetric central configurations of the 5-body problem with four masses equal to 1 varying the fifth mass 1/m from 1 to infinity. This explains why the work of this paper completes the study of Alvarez-Ramírez et al (2008).…”

“…Note that the study in Alvarez-Ramírez et al (2008) is equivalent to study the symmetric central configurations of the 5-body problem with four masses equal to 1 varying the fifth mass 1/m from 1 to infinity. This explains why the work of this paper completes the study of Alvarez-Ramírez et al (2008). Thus the work done in Alvarez-Ramírez et al (2008) and the one in the present paper provide a skeleton of the families of symmetric central configurations of the spatial 5-body problem with four equal masses and their bifurcations.…”

“…2), and they conjectured that these are all the central configurations of the spatial 5-body problem with equal masses. Alvarez-Ramírez et al (2008) proved that if there are other central configurations of the 5-body problem with equal masses they must be non-symmetric. Lee and Santoprete (2009) used a computer algebra algorithm, which is devised to find all the isolated solutions of a polynomial system, to find all the isolated central configurations of the 5-body problem with equal masses.…”